How to check if $x^{21} + 2x^8 + 1$ and $x^{21} + 2x^9 + 1$ are irreducible in $\mathbb{Z}_3$?

Question

The full question is to answer if these two polynomials have multiple zeros. There is a theorem in my book that says that if $f(x) \in F[x]$ and $\operatorname{char}(F) = p ≠ 0$ then $f(x)$ has multiple zeros only if $f(x) = g(x^p)$ for some $g(x) \in F[x]$.

This last condition is easy to check for, since the exponents of the first polynomial have $gcd(21, 8) = 1$ while the second polynomial has $gcd(21, 9) = 3$ therefore the second polynomial should have a multiple zero but only if it is irreducible.

How can I check if either of these polynomials are irreducible?

Answer 1

I ended up using Wolfram Alpha to factor the polynomials:

For $x^{21} + 2x^8 + 1$

http://www.wolframalpha.com/input/?i=factor+x%5E21+%2B+2x%5E8+%2B+1

For $x^{21} + 2x^9 + 1$

http://www.wolframalpha.com/input/?i=factor+x%5E21+%2B+2x%5E9+%2B+1

For both pages, scroll to the bottom to find the factorization in $GF_3$